Annex X.21 –
DEFORMATION OF THE NATURAL MEDIA

The deformation of the natural media
(NM - gases, liquids, solids etc.) under the action of external
forces is a very vast subject, which is approached starting with the
physics textbooks and continuing with the specialized studies focused
on fields such as statics and fluid dynamics, strength of materials
and many others.

This annex will show only few of the
simple relations which determines the attributes values specific to a
deformation (a state change), depending on the attributes of the
agent which generates it. If we have a medium portion with a volume V
(for example, with a spherical shape), by an application on the
body surface of an even pressure p, a ΔV volume
decrease is obtained. The ratio
is the relative volume variation (variation of the
volume unit). If the variations are elementary, the following amount
is being defined:

(X.21.1)

named compressibility. The
reverse amount of the compressibility is the elasticity modulus E.
The dimensions of E are the ones specific to a pressure. If
the volume variation takes place as a result of a temperature
variation, an isobar dilatation coefficient shall be analogously
defined:

(X.21.2)

As for the solids, an amount is
defined, being named unitary strain:

(X.21.3)

with the components
(normal unitary
strain) according to the direction of
the normal’s versor ,
and (unitary tangential
strain) into the plane of the element
dA.
Between the amounts E
(longitudinal elasticity modulus),
G (transverse elasticity modulus)
and μ (Poisson’s coefficient or
transverse contraction coefficient)
there is the following relation:

(X.21.4)

Comment
X.21.1: Attention! Do not mistake the symbol G for the transverse
elasticity modulus with the notation for the G-type media class. The
notion of normal unitary strain
for S-type media is identical with the pressure one from the L or
G-type media, involving the normal component of the variation of an
energetic flux (a force) applied on a surface with an area A.
According to the objectual philosophy, once with the introduction of
the concept of real bounding surface
(RBS) of a MS, in which the tangential flux components occur, the
notion of tangential strain regains its original meaning, similar
with the concept of normal effort, which is also a pressure, but this
time, it occurs on the cross section of a RBS. Otherwise speaking,
the tangential strain is not applied into the plane of the element
dA, but on
the element of the transverse section of RBS. According to the
classic approach (through abstract surfaces), it is clear that the
tangential strain could not be regarded as a pressure, because the
normal area on the tangential strain was null. As regards the term of
transverse contraction,
another remark needs to be done. First of all, this term has a clear
meaning for the stretching strains of the solid or liquid materials,
and is caused by the property of these media to preserve their
volume; consequently, an increase of a dimension (by means of
traction) generates a decrease (contraction) of the cross section
dimensions. In case of the compression strains, there is an obvious
increase of the cross section sizes, therefore, a transverse
dilatation occurs, rather than a contraction. If we shall consider
the dilatation as a negative contraction (#), then, the term can
still be used.

If there is a mean
unitary strain
and the mean specific elongation ,
then, the following relation is applicable:

(X.21.5)

The amounts
,
,
represent specific
elongations (as compared to the axes
X,Y,Z) and
the amounts ,
and are angular deformations or specific
slides against that axis as well. In
case of the isotropic bodies, we have:

,
,
(X.21.6)

and:

,
,
(X.21.7)

or vice versa, the relations between
the unitary strains and the specific deformations:

,
,
(X.21.8)

and:

,
,
(X.21.9)

The specific deformation’s
potential energy (the energy stored into the space unit) is:

(X.21.10)

and if the relations between the
specific distortions are linear, the relation X.21.10 becomes:

(X.21.11)

The same energy may be written
depending only on the unitary strains:

(X.21.12)

or on the specific deformations:

(X.21.13)

All these complicated and probably
boring relations were mentioned with a clear purpose, namely, to
underline the existence of a deformation of the material media
depending on the action of some forces (of some energy fluxes). All
these deformations have a common feature - they are proportional with
the applied force, and they are a state attribute of the potential
energy stored in that medium. Consequently, the objectual philosophy
asserts that:

There cannot exist a material
medium which is non-deformable.

The above mentioned statement, with a
particular case, the non-existence of the incompressible media112,
can be demonstrated by the reduction ad absurdum method. If we
are assuming that a non-deformable medium would exist, this means
that it would have a null 
compressivity according to the relation X.21.1, or accordingly, the
infinite elasticity E modulus. This fact would attract on the
one hand an infinite propagation velocity of the compression waves,
and on the other hand, the impossibility of existing such waves
because their specific local compression feature would not be able to
exist. Furthermore, the potential energy cannot be stored and
restored in a non-deformable medium, because the external state
attribute for this kind of energy is even the medium deformation, as
we have mentioned in chapter 7 and in the relations X.21.10…X.21.13.

The deformability of a specific medium
depends on the type of the medium’s elements, on the bonds type
(interactions) between the elements, on the intensity, temporal
distribution and anisotropy of these bonds. The most non-deformable
(more rigid) media known so far (such as the diamond or some carbide
types) however have finite elasticity modules and finite propagation
velocity rates of the pressure waves.

112The media compressibility is let aside in
some papers (being considered as incompressible), but this aspect
aims only to simplify the relations deployed in case of some
processes, in which the compressibility is not important (such as,
for instance, the motion of some bodies with low velocity through
the fluid media).